You could also study the Lam Rim Chenmo by Tsongkhapa, which is an excellent translation, or the many books by HHDL.
My Gelug teachers are also less cautious about giving initiations into the Kriya/Action class of tantra as these do not for the most part carry tantric vows or daily
recitation commitments (though some teachers may impose a mantra commitment or retreat in rare cases). Practices such as Medicine Buddha, Tara and Chenrezig are accessible to
those who are ready to generate the Bodhicitta attitude and embark on the Mahayana path.
Practices of the Anuttarayoga/Highest Yoga class of tantra are more restricted (the common ones in the Gelug tradition are Yamantaka, Guyasamaja, Chakrasamvara, Vajrayogini and Chittamani Tara though there are others). All of my teachers are quite cautious about these, as they involve daily recitation commitments in many cases of both 6 Session Guru Yoga and the practice of a deity sadhana. I have also attended initiations where the lama imposed a retreat commitment. Having taken the tantric vows one has a very strong and binding relationship with a guru as well. So I always advise caution before taking these types of initiations. (With the exception of Kalachakra, which can be taken as a blessing).
I remember His Holiness the Dalai Lama indicating during a Kalachakra initiation in India that those who could access live streaming in Tibet could consider that they had received the initiation, but I don't know if this is a special case because of the open nature of the Kalachakra teachings. I also know the issue of transmission over the net, even live streaming, is one that many lamas have varying opinions on.
A foolish man proclaims his qualifications,
A wise man keeps them secret within.
A straw floats on the surface of water,
But a precious gem placed upon it sinks to the depths